On the interpretation of s(r) information using SF s QTAIM based descriptor:

The relatively simple case of water in its ³B1 state (Fig.1) is chosen as an example of application of QTAIM based descriptor SF_s with the aim to analyze whether such tool enables one to gain interesting and valuable insights regarding the transmission of electron spin density information and the magnetic coupling mechanism (ferromagnetic or anti-ferromagnetic coupling between atoms, spin exchange or super-exchange etc.) that are behind such transmission information. The interpretation of the results obtained in water triplet are very encouraging and, of course, pushed us to extend the application of the SF_s chemical descriptor to molecular systems that are much complicated with respect this first “simpler” case. Nevertheless the results described till now for the water system do not explain in a very exhaustive way, which are the mechanism behind the transmission of electron spin density information. In particular, it is not clear if s(r) is transmitted

through spin delocalization or spin polarization mechanisms, or also through their in tandem operation. In a very recent work by Deutch et al, the experimental decomposition of the electron density in its spin counterparts is performed for an azido double bridged Cu-Cu molecular system (Figure 7) using an extended version of the well known Hansen&Coppens multipolar model^[5], that permits to refine data-set of very good quality obtained combining both polarized X-ray and neutron diffraction techniques.^h

Fig.7: azido double-bridge Copper II di-nuclear complex; the azido groups bridge the two Cu(II) ions through two terminal N atoms (µ-1,3), in what is called an END-TO-END coordination mode (EE).

(a) (b)

Fig.8: separated α (a) and β (b) electron density distributions of azido double bridge di-nuclear (Cu-Cu) complex obtained combining Polarized Neutron Diffraction and X-Ray Diffraction experiments, using a spin-split version of the

original Hansen&Coppens Multipolar Model refinement

In their work the authors discuss the electron spin density distribution in terms of orbital interactions; in particular they use a fragment orbital approach and consider the interactions between the highest occupied d orbitals of copper atoms and the two (one for each azido bridge) highest doubly occupied πgerade azido orbitals. The interpretation of s(r) distribution was then done through the concept of spin delocalization (due to the overlap between the fragment orbitals) and

h the application of the topological descriptor SF to this class of complexes will be better described in next sub-sections

spin polarization (that involve both the πgerade and the lowest unoccupied πungerade molecular orbitals of the N₃^- fragment and where the fundamental importance of π-π* excitations to produce the ferromagnetic coupling between the two copper atoms has been emphasized); to this aim the authors have employed either two-electron active orbitals models either more complicated ones, i.e.

multi-electron models using more sophisticated multi-configurations wavefunction models.

Mimicking the orbital interpretation adopted by Aronica et al, we thought it worth introducing in our SF/SF_s analysis a physically-rooted partitioning of the values of the observables listed in Tab.1 (and also of their derived SF and SF_S values) in terms of a sum of two contributions: a magnetic one arising from the unpaired α-electrons orbitals (hereinafter magnetic orbitals) and a reaction or relaxation contribution due to the remaining orbitals.^[26]

Computational details:

The correct analysis of the decomposition of both ρ(r) and s(r) in an open shell molecular system as

3B1 water triplet requires the use of some particular precaution in the calculation of the wave-function^i[19]. For this reason, different levels of theory were employed during all the in vacuo quantum mechanical simulation. Thus we performed CASSCF(8,8), UHF (Unrestricted Hartee Fock), ROHF (Restricted Open Hartree Fock) calculations with a 6–311++G(2d,2p) basis set;

moreover computations on both spin-contamination annihilated and spin contaminated UHF wave-functions were performed; such calculations revealed that spin contamination by states of higher multiplicity than the triplet state was very small (<S²>=2.0069), and becomes almost negligible when annihilation procedure is applied ( <S²>=2.000014). Static electron correlation corrections were considered by performing a CASSCF(8,8) computation. To this aim the starting guess was taken from the UHF spin contamination annihilated Natural Orbitals, obtaining a Slater determinant expansion of the wavefunction which included 3136 configurations of the correct symmetry and spin multiplicity. Thanks to the Natural orbitals analysis^j magnetic orbitals were very easily singled out, based on their occupation numbers, in all cases. In ROHF calculations, the wavefunction include natural orbitals with occupation numbers equal to one by definition because both β-density and relaxation contribution are equal to zero everywhere; for the other adopetd levels of theory the occupation numbers of magneric orbitals were either one or marginally different from one (highest deviation from one being 0.0003 for CASSCF(8,8) wavefunction). Spin densities were instead

i in particular we paid particular attention on the problem of spin contamination and static and dynamic electron correlation

j pop= no option in G09 program package

calculated from the naturals orbitals obtained from separate diagonalizations of the α- and β-density matrices^k.

Results and discussion:

In the molecular system ³B₁ H₂O the two magnetic orbitals have B₁ and A₁ symmetry. They are obtained through the diagonalization of the first order density matrix and by taking those natural orbitals with occupation number (n) equal to or marginally different from one.

Fig.9: 3D spin density plots in the (x,y) and (z,y) plane, as evaluated just for the B₁ and A₁

symmetry magnetic natural orbitals at the CASSCF(8,8) level of theory. An isosurface value of 0.015 a.u.

was selected, with maxima of spin density equal to 0.596 a.u. for B₁ symmetry orbital and 0.250 a.u. for A₁ symmetry orbital.

Figure 9, 10 and 11 report the 3D plot of the two magnetic natural orbitals (NOs) densities, of their sum and of the total spin density, respectively, for the CASSCF(8,8) level of theory wavefunction.

For magnetic orbitals, ρ(r) ≡ s(r), ∇²ρ(r) ≡ ∇²s(r) , ρα(r) ≡ s(r), ∇²ρα(r) ≡ ∇²s(r) while ρβ(r) and

∇²ρβ(r) are both null, so that only s(r) and ∇²s(r) values need to be reported (Table 6, values in parentheses). It is very important to stress that s(r) and ∇²s(r) contributions due to the remaining orbitals are obtained by subtracting those of the magnetic orbitals from the total s(r) and ∇²s(r) values. Their contributions may differ from zero at a given point, despite they are both null when integrated over the whole space.

k pop=noab option G09 program package; For CASSCF method, G09 apparently doesn’t calculate and save spin density information. To this aim the IOP(5/72=1) option is mandatory, furthermore at the bottom of input file before the name selected for the .wfn file a “1 1” string needs ro be introduced. Finally, SlaterDet option should be used in this case in the CASSCF calculation. In this way is possible to recover a correct α-density through the pop=noa option (but not the correct spin density through pop=noab, nor the correct β-density through pop=nob). From the total density and the α-density the electron spin density and electron spin density Laplacian is obtained by difference: s(r) = 2ρ_α(r) -ρ(r);

∇²s(r) = 2∇²ρ(r) -∇²ρ(r).

Fig.10: Same as Fig.9 above, but summing up the spin density contributions of the B₁ and A₁ symmetry magnetic natural orbitals. Maxima of spin density fall at 0.603 a.u.

Fig.11: As Figures 9 and 10 above but plotting the total spin density. The maxima and minima of spin density fall at 0.618 a.u and -0.008 a.u. respectively.

As already discussed in the previous paragraph, besides the (3,–1) bond critical point (bcp) of the ρ(r) distribution (bcp 1, Fig. 1), suitable references points (rps) of the -∇²ρ(r) = L(r) field for the SF analysis have been selected (Fig. 1).

Table 6 reports the values of ρ(r), ρα(r), ρβ(r), s(r) and the corresponding Laplacians at each reference point mentioned; such results have been obtained using wavefunctions evaluated at a common geometry (the UHF/6–311++G(2d,2p) optimized geometry^l). The locations of each reference point differ as they correspond to the selected critical point for the considered wavefunction; however since each critical point comes from the analysis of ρ(r), they almost coincide for the three computational levels shown in Table 6.

l In case of theUHF/6-311++G(2d,2p) level of theory we refer to the spin-contamination annihilated wavefunction

RP ρ(r) ∇²ρ(r) s(r) ∇²s(r) ρα(r) ∇²ρα(r) ρβ(r) ∇²ρβ(r) CASSCF(8.8)//UHF(6–311++G(2d.2p))

1 0.291 -2.06 -0.0075 (0.0020) 0.24 (0.13) 0.142 -0.91 0.149 -1.15 2 0.888 -5.08 0.0763 (0.0508) 0.90 (1.21) 0.482 -2.09 0.406 -2.99 3 1.022 -6.64 0.0219 (0.0038) 1.73 (1.97) 0.522 -2.46 0.500 -4.18 4 0.614 -1.23 0.3824 (0.3722) -4.45 (-4.40) 0.498 -2.84 0.116 1.61

UHF/(6–311++G(2d.2p)) spin contamination annihilated wavefunction

1 0.288 -2.14 -0.0050 (0.0029) 0.21 (0.11) 0.141 -0.96 0.146 -1.18 2 0.888 -5.17 0.0631 (0.0511) 1.07 (1.18) 0.475 -2.05 0.412 -3.12 3 1.030 -6.85 0.0051 (0.0037) 2.04 (1.95) 0.518 -2.40 0.513 -4.45 4 0.610 -1.18 0.3818 (0.3677) -4.54 (-4.34) 0.496 -2.86 0.114 1.68

ROHF//UHF(6–311++G(2d.2p))

1 0.287 -2.14 0.0031 0.11 0.145 -1.01 0.142 -1.13 2 0.890 -5.21 0.0483 1.20 0.469 -2.01 0.421 -3.20 3 1.031 -6.87 0.0032 1.95 0.517 -2.46 0.514 -4.41

4 0.607 -1.13 0.3637 -4.28 0.485 -2.7 0.121 1.57

Tab.6: Values of electron density, electron spin density, Laplacian of total ρ(r), Laplacian of the α and β counterparts of ρ(r) and Laplacian of spin density distribution (in a.u.) at each critical point considered in Fig.1 for the three adopted computational levels of theory; in parentheses the contributions from the two magnetic NOs are reported. For these NOs ρ(r) ≡ s(r), ∇²ρ(r) ≡ ∇²s(r) , ρ_α(r) ≡ s(r), ∇²ρα(r) ≡ ∇²s(r) while ρ_β(r) and ∇²ρβ(r) are both null; in the specific case of

the ROHF wavefunction, s(r) ≡ ρ_α,mag(r) and ∇²s(r) ≡ ∇²ρα,mag(r) where ρ_α,mag(r) and ∇²ρα, mag(r) denote the magnetic contribution to ρ_α(r) and ∇²ρα(r), respectively.

The decomposition of ρ(r) and s(r) in contributions given by the two magnetic orbitals and the reaction orbitals show how the former dominate both the large s(r) and its largely negative ∇²s(r) at the two symmetric (3,+1) L(r) points 4 and 4’ as well as the the spin density depletion (∇²s > 0) at the in-plane NBCC 3 associated to the lone pair (see Tab.6). At bcp 1 in the case of the CASSCF(8,8) and UHF level of theory, the remaining orbitals overreact to the small positive s(r) contribution due to the two magnetic orbitals.

Fig.12: Electron density Laplacian, electron spin density and its Laplacian in the (y,z) plane for ³B₁ H₂O, at (top) CASSCF(8,8) and (bottom) UHF/UHF spin-contamination annihilated computational levels. Atomic units (a.u.) are used throughout. Contour maps are drawn at interval of ±(2,4,8)⋅10ⁿ, –4 ≤ n ≤ 0 (s(r), ∇²s(r)) and –3 ≤ n ≤ 0 (∇²ρ(r)).

Dotted blue (full red) lines indicate negative (positive) values and full black lines mark boundaries of atomic basins.

The O–H bond critical point (bcp, 1) and the bonded charge concentration point (BCC, 2) are shown as black and green dots, respectively.

Fig.13: Electron density Laplacian, spin density and its Laplacian in the (x,z) plane, at (top) CASSCF(8,8) and (bottom) UHF/UHF spin contamination annihilated computational levels. Contour levels as in Figure 12. The non-bonded charge

concentration (NBCC, 3) and the (3,+1) L(r) rcps (4) are shown as green and red dots, respectively.

The last consideration is not true in the case of the ROHF wavefunction, because the reaction mechanism is unattainable and, as a consequence, s(r) remains positive at this CP. Considering the bonded charge concentration (BCC 2, coloured green), the contributions to s(r) from the two set of orbitals are equal in sign and definitely larger for the magnetic orbital set, but the ∇²s(r) value of

the magnetic orbitals is positive (∇²s = 1.2 au) and larger in magnitude than that of the remaining orbitals which is negative (∇²s = –0.3 au and -0.1 au for the CASSCF(8,8) and the UHF wavefunctions, respectively). This leads to a global dilution of the spin density at BCC 2.

Fig.14: Electron density Laplacian, electron spin density and its Laplacian in the (y,z) plane for ³B₁ H₂O due just to the non-magnetic natural orbitals for the CASSCF(8,8), the UHF contamination annihilated and the UHF spin-contaminated computational levels. Atomic units (a.u.) are used throughout. Contour levels as in Fig.12. The O–H bond

critical point (bcp, 1) and the bonded charge concentration point (BCC, 2) are shown as black and green dots.

Considering static and dynamic electron correlation at the CASSCF(8,8) level of theory, one may generally observe (Tab. 6, Figures 12-14) a similar qualitative picture relative to that at the UHF spin-contamination annihilated level; this agreement increases a lot when just contributions given by the magnetic orbitals are compared (Tab. 6). The spin density at the in-plane NBCC 3 associated to the lone pair shows a completely different behaviour. In fact the introduction of electron correlation effects raises s(r) by more than five time, with respect to the value of the spin-contamination annihilated wavefunction; this increase in the s(r) value is due the reaction or relaxation contribution (Tab. 6). This noticeable effect due to electron correlation can be also observed in the s(r) maps reported in Fig.13, where the small region of negative spin density of the UHF model lying close to the non bonded maximum disappears in the corresponding CASSCF(8,8) plot. The effects of electron correlation are even more evident if the UHF model spin contamination

electron correlation effects involve the reaction orbitals, as it is possible to deduct from Fig. 14 and Fig. 15 where maps of s(r) and ∇²s(r) relative to the planes shown in Fig. 12-13 and obtained using only these natural orbitals are reported.

It is now interesting to comment briefly on the different portraits of the ED and of the electron spin density Laplacians. In water, ∇²ρ(r) implies relatively contracted valence shell charge concentration (VSCC) zones, mainly localized around nuclei and along covalent bonds, while the

∇²s(r) negative regions are definitely more extended and possibly disjoint (Fig. 12 and 13).

Furthermore, a given region of space may occur to be diluted for ρ(r) and concentrated for s(r) or vice-versa.

Fig.15: As in Fig.14 above, in the (x,z) plane with same contour levels. The non-bonded charge concentration (NBCC 3) and the (3,+1) L(r) critical points (CPs 4 and 4’) are shown as green and red dots.